Proposition 20

The mapping $x\mapsto$ exp(2$i\pi \lambda (x)$) is a unitary character of $\mathbf{Q}_p$ whose kernel is $\mathbf{Z}_p$.

For $x_1$ and $x_2$ in $\mathbf{Q}_p$, one has by definition $\lambda (x_1+x_2)-\lambda (x_1)-\lambda (x_2)\in$ $\mathbf{Z}_p\cap \mathbf{Q}=\mathbf{Z}$. The mapping $\lambda$ is moreover locally constant since $\lambda (x+y) =\lambda (x)$ if $y\in \mathbf{Z}_p$. It then follows that $x\mapsto$ exp(2$i\pi \lambda (x)$) is a unitary character of $\mathbf{Q}_p$. Since $\lambda (x)\in \mathbf{Z}$ if and only if $x\in \mathbf{Z}_p$, the kernel of this character is $\mathbf{Z}_p$.

We recall that the normalised Haar measure on the additive group of $\mathbf{Q}_p$ is the unique Haar measure $\mu$ such that $\mu(\mathbf{Z}_p) = 1$ (INT, VII, §1, No.$^o6$, example).